Can the old and new machines share email messages if one is running Office 2007 and the other Office 365? Or do I have to upgrade the old machine to Office 365?
Yes, but this is more dependent on the protocol you use to access the messages than the versions of Office.
The old machine is currently set up as POP3. Is that going to be a problem?
Most likely yes. Post Office Protocol (POP) is "first generation" protocol for dealing with e-mail before the age when someone was likely to be trying to juggle the same e-mail messages on multiple devices. It is a client side arrangement, which means your e-mail messages (in most default configurations) are downloaded to the device you're reading them on and as soon as that's done, or at some set time afterward, they are deleted from the server and are no longer available for additional download to other devices.
If you expect to be wanting to juggle the same messages across multiple devices you should convert to the Internet Message Access Protocol (IMAP). IMAP is a server side protocol, where all messages, and the folders that you set up, and the filters you create to sort them into those folders, are all created and stored on the e-mail server. The devices that access your mail all connect to the server and get the folder setup and the message headers (usually) until you open them for reading, at which time the message body is downloaded, but not deleted from the server.
Is there a way that I can "ignore" emails that I see on one mobile device and have it also not show up on any other mobile devices, but not be deleted and still show up on the PC?
Not really, at least not if you're using deletion to do it. You could create a folder on the e-mail server named, say, Reviewed, and when you read a message on the mobile device and don't want it showing up in the Inbox on the others moving it to Reviewed when you are done. It will be available, everywhere, in "Reviewed" and gone everywhere from your inbox.
The entire concept of IMAP is to make everything available, in precisely the same way, on every device that accesses the server for a specific e-mail account.
If I set up an email folder structure, will that structure be mirrored on the server and also on all of the other devices? If I move an email from one folder to another or move an entire folder, will that be propagated to the server and all devices?
Yes, as I described above. That's one of the primary reasons IMAP access came into existence and has largely supplanted POP as the access protocol of choice.
Are any of the books out there any good at helping me sort all this out? If so, which ones?
Not something I can give you a recommendation for. I hope the above clears up at least 90% of your questions.
Edited by britechguy, 18 June 2017 - 08:21 PM.
Brian AKA Bri the Tech Guy (website in my user profile) - Windows 10 Home, 64-Bit, Version 1803, Build 17134
. . . the presumption of innocence, while essential in the legal realm, does not mean the elimination of common sense outside it. The willing suspension of disbelief has its limits, or should.
~ Ruth Marcus, November 10, 2017, in Washington Post article, Bannon is right: It’s no coincidence The Post broke the Moore story